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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrssre | Structured version Visualization version GIF version |
Description: A subset of extended reals that does not contain +∞ and -∞ is a subset of the reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
xrssre.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
xrssre.2 | ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) |
xrssre.3 | ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) |
Ref | Expression |
---|---|
xrssre | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrssre.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) | |
2 | ssxr 11290 | . . . . 5 ⊢ (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) |
4 | 3orass 1089 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (𝐴 ⊆ ℝ ∨ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))) | |
5 | 3, 4 | sylib 217 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∨ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))) |
6 | 5 | orcomd 868 | . 2 ⊢ (𝜑 → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ)) |
7 | xrssre.2 | . . . 4 ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) | |
8 | xrssre.3 | . . . 4 ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) | |
9 | 7, 8 | jca 511 | . . 3 ⊢ (𝜑 → (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴)) |
10 | ioran 981 | . . 3 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴)) | |
11 | 9, 10 | sylibr 233 | . 2 ⊢ (𝜑 → ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) |
12 | df-or 845 | . . 3 ⊢ (((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ) ↔ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ)) | |
13 | 12 | biimpi 215 | . 2 ⊢ (((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ)) |
14 | 6, 11, 13 | sylc 65 | 1 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 ∨ w3o 1085 ∈ wcel 2105 ⊆ wss 3948 ℝcr 11115 +∞cpnf 11252 -∞cmnf 11253 ℝ*cxr 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 |
This theorem is referenced by: supminfxr2 44637 |
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